Now let us consider the construction for a surface of genus \(g\)  and
\(r+1\)  boundary components.
This construction creates a 2-form on
the internal fusion \({\bf D}(G) \,=\, G \times G \)  as ([1]})
\( \omega \, =\, \frac{1}{2} ( a^* \theta ,\, b^* \overline{\theta }) + \frac{1}{2} ( a^* \overline{\theta },\, b^* {\theta })+\frac{1}{2} ( (ab)^* \theta ,\, (a^{-1}b^{-1})^* \overline{\theta }). \) 
